By Lemma REF , for any \(v^{\prime } \in V(S^{\prime })\) , dim \(\tau ^{\prime } = \text{dim }h(\tau ^{\prime }_{v^{\prime }}) = n-1\) . Hence, for \(t \in \tau ^{\prime }\) , \(h_t(v^{\prime })\)  uniquely determines the tropical map \(h_t: \Gamma _t^{\prime } \rightarrow \Sigma (X)\) , i.e., \(ev_{v^{\prime }}:\tau ^{\prime } \rightarrow \sigma (v^{\prime })\)  is injective. After recalling Proposition \(3.23\)  of [1]}, note that since we assumed \(\tau \)  is marked by the realizable tropical type \(\tau ^{\prime }\) , we may identify \(\tau ^{\prime }\)  with a face of \(\tau \) . Let \(c: \Gamma (G^{\prime },l) \rightarrow \Gamma (G,l)\)  be the morphism of universal families associated with the face inclusion \(\tau ^{\prime } \subset \tau \) , and a corresponding graph contraction \(\phi : G \rightarrow G^{\prime }\) . Note that \(c\)  maps the cones corresponding to vertices and edges in the spine \(S^{\prime }\)  of \(G^{\prime }\)  to the cones of the spine \(S\)  of \(G\) . In particular, the contraction map \(\phi : G \rightarrow G^{\prime }\)  restricts to a morphism \(\phi |_{S}: S \rightarrow S^{\prime }\) . We label the corresponding subcomplexes of \(\Gamma (G^{\prime },l^{\prime })\)  and \(\Gamma (G,l)\)  as \(\Gamma (S^{\prime },l)\)  and \(\Gamma (S,l)\)  respectively. Let \(v \in V(S)\) , and note by Lemma REF  that \(\sigma (\phi (v)) \in {P}\)  and dim \(h^{\prime }(\tau ^{\prime }_{\phi (v)}) = n-1\) . Hence dim \(h(\tau _v) \ge n-1\) .
